About the definition of the norm $||A||$ of a linear operator $A$ in "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

Question

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

For a linear operator $A \in L(\mathbb{R}^n, \mathbb{R}^m)$, Rudin defined the norm $||A||$ of $A$ as follows:

$||A|| := \sup_{|x| \leq 1} |Ax|$.

Because I am new to the norm of a linear operator, I feel the following definition is more natural:

$||A|| := \sqrt{\sum_{i=1}^{n} |A(e_i)|^2}$.

Can we replace $||A|| := \sup_{|x| \leq 1} |Ax|$ with $||A|| := \sqrt{\sum_{i=1}^{n} |A(e_i)|^2}$ in "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin?